How do the current input and the output of the previous time step get combined in an LSTM?
(1) $X_t$ and $H_{t-1}$ are concatenated. The blog you cited explained its notation "Lines merging denote concatenation". For example, if $X_t=[1,2,3]$ and $H_{t-1}=[4,5,6,7]$, then their concatenation is $[1,2,3,4,5,6,7]$

(2) When you say "input weights" or "weights of the input of the previous time step", are you referring to the $W_i$ in your cited blog? If so they are not the weights of the connections between the time-steps/units. They are part of the input gate only. The connections between the time-steps/units do not have weights applied to them.